Rational functions of Degree Four that Permute the Projective Line over a Finite Field

Abstract

Recently, rational functions of degree three that permute the projective line $\Bbb P^1(\Bbb F_q)$ over a finite field $ \Bbb F_q$ were determined by Ferraguti and Micheli. In the present paper, using a different method, we determine all rational functions of degree four that permute the $\Bbb P^1(\Bbb F_q)$.